PROCEEDINGS
OF THE TOMSK POLYTECHNIC INSTITUTE
OF THE ORDER OF THE OCTOBER REVOLUTION AND THE ORDER OF THE RED BANNER OF LABOR
NAMED AFTER S. M. KIROV

Volume 295                    1976

SELECTION OF PARAMETERS FOR LONGITUDINAL AND TRANSVERSE COMPENSATION INSTALLATIONS IN LONG-DISTANCE POWER TRANSMISSION LINES WITH INTERMEDIATE SYSTEMS

R. I. BORISOV, I. A. BEZLER, V. I. GOTMAN

(Presented by the scientific seminar of the Department of Electrical Systems and Networks)

Of the many means for increasing the transmission capacity and stability of long-distance power transmission, longitudinal capacitive compensation of the inductance of an AC power transmission line is one of the principal means in terms of technical and economic conditions [1, 2].

The question of selecting the parameters of compensating devices (CD) for a specified transmission capacity under the conditions of ensuring static stability and the laws of voltage distribution in a long-distance power transmission line (LDPTL) without intermediate systems (IS) was solved in [1], and in power transmissions with IS the question of selecting CD parameters was partially solved in [3, 4]. In view of the complexity of the structure of the IS, the use of various types of transverse compensation installations (TCI), and depending on the position of the IS along the route of the long-distance power transmission, as well as on the point of connection of the IS and TCI (to the inputs or outputs of longitudinal compensation installations), the problem of selecting CD parameters for an LDPTL with IS according to limiting regimes and static-stability conditions presents serious difficulties.

To improve the dynamics of the LDPTL, transverse controllable CD are required. Such controllable CD may be controlled reactors, synchronous and static compensators. By a static compensator is meant a controlled reactor and uncontrolled static capacitors connected in parallel with it. At the receiving and sending ends of the power system the CD are selected from no-load conditions, and in power transmissions with longitudinal capacitive compensation installations (LCCI) and IS—additionally according to the conditions for limiting the voltage on the plates of the LCCI capacitors in all regimes, taking into account the limit of the IS regime with respect to reactive power.

Below a method is given for calculating the parameters of the LCCI and TCI when the IS is connected to the inputs or outputs of the LCCI at various distances from sending system No. 1, for a specified transmission capacity of the LDPTL from the conditions of system stability and taking into account the permissible voltage level at the controlled nodes. To simplify the calculations, the terminal CD of sending power system No. 1 and receiving power system No. 2 are introduced into the constants of the four-terminal networks \([A_1]\) and \([A_2]\).

The derivation of equations for determining the CD parameters is carried out with reference to Fig. 1, which represents the LDPTL, IS, LCCI and TCI. If the TCI is taken as controllable, then it appears in the circuit as if a second IS. The total length of the LDPTL is 1000 km and the quantities \([A_1]\ne[A_2]\),

and the active powers are also specified, with \(P_1 \ne P_2\). In what follows we shall assume that, in the general case, \(U_1 \ne U_2 \ne \mathrm{const}\): the first because of “strong-action” regulators, the second because the receiving system is of infinite power. We shall regard the value of the slope coefficient of the intermediate system (IS), \(q_{\mathrm{ps}}\), as a known quantity, since the specific IS circuit is known. When the IS is connected to the terminals of the transverse compensation installation (TCI), we shall not include the additional reactive power \(Q_{p4}\) in the calculation, since the IS itself provides the reactive-power balance for voltage regulation at node \(U_6\).

a)

b)

Fig. 1. Scheme of a compensated long-distance power transmission line:
a) the IS is connected to the terminals of the series compensation installation; b) the IS is connected to the inputs of the series compensation installation.

For the circuit under consideration, the equivalent matrix expression, which includes the left-hand part of the long-distance electric transmission line (LETL), the TCI, and the series compensation installation (SCI), takes the form

\[ [\dot A_{\mathrm{e}}]=[\dot A_1][\dot A_3][\dot A_k], \tag{1} \]

or, in expanded form, taking into account that

\[ \dot B_3=\dot C_k=0 \quad \text{and} \quad \dot A_3=\dot D_3=\dot A_k=\dot D_k=1, \]

\[ \dot A_{\mathrm{e}}=\dot A_1+\dot B_1\dot C_3; \tag{2} \]

\[ \dot B_{\mathrm{e}}=\dot A_1\dot B_k+\dot B_1\dot C_3\dot B_k+\dot B_1; \tag{3} \]

\[ \dot C_e=\dot C_1+\dot D_1\cdot \dot C_3; \tag{4} \]

\[ \dot D_e=\dot C_1\dot B_k+\dot D_1\cdot \dot C_3\cdot \dot B_k+\dot D_1. \tag{5} \]

The system of equations for the active and reactive powers at the ends of the sections of the long-distance electric transmission line, according to the directions adopted in Fig. 1, is written in general form as

\[ P_i=\frac{U_i\cdot U_j}{B_i}\sin(\delta_i-\delta_j); \tag{6} \]

\[ Q_{13}=\frac{U_1^2D_e}{B_e}-\sqrt{\left(\frac{U_1\cdot U_6}{B_e}\right)^2-P_1^2}; \tag{7} \]

\[ Q'_{13}=-\frac{U_6^2A_e}{B_e}+\sqrt{\left(\frac{U_1\cdot U_6}{B_e}\right)^2-P_1^2}; \tag{8} \]

\[ Q'_2=-\frac{U_6^2D_2}{B_2}+\sqrt{\left(\frac{U_2\cdot U_6}{B_2}\right)^2-P_2^2}; \tag{9} \]

\[ Q_2=-\frac{U_2^2A_2}{B_2}+\sqrt{\left(\frac{U_2\cdot U_6}{B_2}\right)^2-P_2^2}. \tag{10} \]

The values of the reactive powers presented in the left-hand sides of equations (7)-(10) are determined from the condition that the voltages on the plates of the static capacitors of the series compensation installation be limited by the balancing reactive powers of sections 1--5 and 6--2. The best utilization of the series compensation installation is achieved when the voltages on both sides of the capacitor banks are equal. This makes it possible to obtain from the series compensation installation the greatest phase-shifting effect. Therefore, from the condition of limiting the voltage at the input of the series compensation installation at node \(U_5\), we determine the required reactive power at the beginning of the long-distance electric transmission from the equation for the left-hand section by the expression

\[ Q_1=\frac{U_1^2D_1}{B_1}-\sqrt{\left(\frac{U_1U_5}{B_1}\right)^2-P_1^2}, \]

taking into account that the voltages entering this equation and equations (6)-(10) must lie within \(U\leq 1.05U_n\).

The equation of the criterion for the limiting operating condition of the system depends on the steepness coefficients and takes the form

\[ q_{61}+q_{62}+q_{pc}=0, \tag{11} \]

where

\[ q_{61}=-\frac{2U_6A_e}{B_e}+\frac{U_1^2U_6}{B_e^2} \left[\left(\frac{U_1U_6}{B_e}\right)^2-P_1^2\right]^{-0.5}; \]

\[ q_{62}=-\frac{2U_6D_2}{B_2}+\frac{U_2^2U_6}{B_2^2} \left[\left(\frac{U_2U_6}{B_2}\right)^2-P_2^2\right]^{-0.5}. \]

Solving equations (1, 7, 11) jointly with respect to \(B_e\), we obtain a nonlinear equation of the form

\[ (F_2^2\cdot P_1^2+F_4^2)\cdot B_e^6 -2F_1\cdot F_2\cdot R_1^2\cdot B_e^5 -(2F_3\cdot F_4-2F_0\cdot F_2\cdot P_1^2- \]

\[ -F_1^2\cdot P_1^2+F_2^2\cdot U_1^2\cdot U_6^2)B_e^4 -(2F_0\cdot F_1\cdot P_1^2-2F_1\cdot F_2\cdot U_1^2\cdot U_6^2)B_e^3- \]

\[ -\left(F_1^2 U_1^2 U_6^2-F_0^2 P_1^2+2F_0F_2U_1^2U_6^2-F_3^2\right)B_3^2+ \]
\[ +2F_0F_1U_1^2U_6^2B_3-F_0^2U_1^2U_6^2=0, \tag{12} \]

where

\[ F_0=U_6U_1^2B_1;\qquad F_1=2U_6U_1^2; \]

\[ F_2=(q_{62}+q_{\mathrm{ps}})(U_1^2D_1-Q_1B_1); \]

\[ F_3=U_1^2U_6\left[Q_1B_1-U_1^2D_1+B_1U_6(q_{62}+q_{\mathrm{ps}})\right]; \]

\[ F_4=(q_{62}+q_{\mathrm{ps}})B_1P_1^2. \]

The obtained equation (12) can be solved by any iterative method.

After determining \(B_3\), one proceeds to finding the coefficients of the quadripole \(A_3, D_3, C_3\), which are equal to

\[ A_3=\frac{U_1^2(B_1-B_3)} {B_3\left(Q_1B_1-U_1^2D_1+B_1\sqrt{\left(\frac{U_1U_6}{B_3}\right)^2-P_1^2}\right)}; \tag{13} \]

\[ C_3=\frac{A_3D_1-1}{B_1}; \tag{14} \qquad D_3=\frac{1+B_3C_3}{A_3}. \tag{15} \]

Solving equations (2–5) and (12–15), we determine the necessary values of series capacitance and shunt compensation, taking into account the limiting nature of the operating condition, i.e., we obtain that

\[ X_k=B_k;\qquad Y_3=\frac{C_3-C_1}{D_1}. \]

When the additional reactive power at the connection node of the intermediate system (IS) \((Q_{p3})\) is taken into account, equation (11) and, consequently, expression (12) will change.

The calculation of the parameters of the compensating devices when the IS is connected to the terminals of the controlled shunt compensation device (Fig. 1b) is carried out analogously to expression (12), except for the changed indices of some components. The selection of the parameters of the compensating devices was also performed for an LPTL with two ISs (the role of the second IS is played by a controlled shunt compensation device).

REFERENCES

1. V. A. Venikov, I. P. Siuda. Calculations of Operating Conditions of Long-Distance AC Power Transmissions. Moscow, “Vysshaya shkola,” 1966.

2. G. E. Pospelov. Elements of Technical-and-Economic Calculations of Electric Power Transmission Systems. Minsk, “Vyshėishaya shkola,” 1967.

3. R. I. Borisov. Selection of optimal parameters of compensating devices for a long line connected with intermediate systems. Transactions of the Siberian Power Engineering Institute, Siberian Branch of the USSR Academy of Sciences, issue 14, 1962.

4. R. I. Borisov, V. I. Gotman. On the synthesis of structures of long-distance power transmissions with intermediate systems. Izvestiya Vuzov, Energetika, 1971, No. 12.